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Theorem ntrclscls00 44079
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that the closure of the empty set is the empty set hold equally. (Contributed by RP, 1-Jun-2021.)
Hypotheses
Ref Expression
ntrcls.o 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
ntrcls.d 𝐷 = (𝑂𝐵)
ntrcls.r (𝜑𝐼𝐷𝐾)
Assertion
Ref Expression
ntrclscls00 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘   𝑗,𝐼,𝑘   𝑗,𝐾,𝑘   𝜑,𝑖,𝑗,𝑘
Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘)   𝐼(𝑖)   𝐾(𝑖)   𝑂(𝑖,𝑗,𝑘)

Proof of Theorem ntrclscls00
StepHypRef Expression
1 ntrcls.o . . . . . 6 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖𝑗))))))
2 ntrcls.d . . . . . 6 𝐷 = (𝑂𝐵)
3 ntrcls.r . . . . . 6 (𝜑𝐼𝐷𝐾)
41, 2, 3ntrclsfv1 44068 . . . . 5 (𝜑 → (𝐷𝐼) = 𝐾)
54fveq1d 6908 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐾‘∅))
62, 3ntrclsbex 44047 . . . . 5 (𝜑𝐵 ∈ V)
71, 2, 3ntrclsiex 44066 . . . . 5 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
8 eqid 2737 . . . . 5 (𝐷𝐼) = (𝐷𝐼)
9 0elpw 5356 . . . . . 6 ∅ ∈ 𝒫 𝐵
109a1i 11 . . . . 5 (𝜑 → ∅ ∈ 𝒫 𝐵)
11 eqid 2737 . . . . 5 ((𝐷𝐼)‘∅) = ((𝐷𝐼)‘∅)
121, 2, 6, 7, 8, 10, 11dssmapfv3d 44032 . . . 4 (𝜑 → ((𝐷𝐼)‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
135, 12eqtr3d 2779 . . 3 (𝜑 → (𝐾‘∅) = (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))))
14 dif0 4378 . . . . . . 7 (𝐵 ∖ ∅) = 𝐵
1514fveq2i 6909 . . . . . 6 (𝐼‘(𝐵 ∖ ∅)) = (𝐼𝐵)
16 id 22 . . . . . 6 ((𝐼𝐵) = 𝐵 → (𝐼𝐵) = 𝐵)
1715, 16eqtrid 2789 . . . . 5 ((𝐼𝐵) = 𝐵 → (𝐼‘(𝐵 ∖ ∅)) = 𝐵)
1817difeq2d 4126 . . . 4 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = (𝐵𝐵))
19 difid 4376 . . . 4 (𝐵𝐵) = ∅
2018, 19eqtrdi 2793 . . 3 ((𝐼𝐵) = 𝐵 → (𝐵 ∖ (𝐼‘(𝐵 ∖ ∅))) = ∅)
2113, 20sylan9eq 2797 . 2 ((𝜑 ∧ (𝐼𝐵) = 𝐵) → (𝐾‘∅) = ∅)
22 pwidg 4620 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ 𝒫 𝐵)
236, 22syl 17 . . . 4 (𝜑𝐵 ∈ 𝒫 𝐵)
241, 2, 3, 23ntrclsfv 44072 . . 3 (𝜑 → (𝐼𝐵) = (𝐵 ∖ (𝐾‘(𝐵𝐵))))
2519fveq2i 6909 . . . . . 6 (𝐾‘(𝐵𝐵)) = (𝐾‘∅)
26 id 22 . . . . . 6 ((𝐾‘∅) = ∅ → (𝐾‘∅) = ∅)
2725, 26eqtrid 2789 . . . . 5 ((𝐾‘∅) = ∅ → (𝐾‘(𝐵𝐵)) = ∅)
2827difeq2d 4126 . . . 4 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = (𝐵 ∖ ∅))
2928, 14eqtrdi 2793 . . 3 ((𝐾‘∅) = ∅ → (𝐵 ∖ (𝐾‘(𝐵𝐵))) = 𝐵)
3024, 29sylan9eq 2797 . 2 ((𝜑 ∧ (𝐾‘∅) = ∅) → (𝐼𝐵) = 𝐵)
3121, 30impbida 801 1 (𝜑 → ((𝐼𝐵) = 𝐵 ↔ (𝐾‘∅) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  c0 4333  𝒫 cpw 4600   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by: (None)
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